Carhart — Direct and Counter Electromotive Forces. 99 



In curve two the efficiency is measured in the same way as 

 before. Again it is seen to increase as the electromotive forces 

 increase. The current, represented by the intercepts of the 

 ordinates between the hyperbola and its asymptote OGr, de- 

 creases as the efficiency increases. The same was true in the 

 other case. 



It remains only to point out that Jacobi's law of maximum 

 rate of working applies only under the condition of a constant 

 B. M. F. in the electric supply. 



This will be evident upon careful inspection of the diagram. 

 The energy spent upon the motor per second is represented by 

 E,(E — E,); the electrical energy supplied by the source per 

 second, by E (E— E,). 



The first expression is the product of the two parts into which 

 the line denoting E is divided at any point by curve I. Since 

 OG bisects the angle between the two rectangular axes, the 

 ordinate of any point of this line equals the abscissa of the 

 points of the hyperbola through which it passes. Thus KQ 

 equals OQ or E, the direct E. M. F. corresponding to the points 

 B and C of the curve. Hence BK equals E— E, ; and the pro- 

 duct of BK and BQ, or the area of the rectangle BMPQ, is the 

 constant EW, of the equation of this hyperbola. The area of 

 all such rectangles is the same for this curve. In a similar way 

 the area of KLPQ is the rate at which electrical energy is sup- 

 plied to the circuit ; and the difference between the two rectan- 

 gles, or the square KLMB, is the heat waste per second at this 

 point of the curve. 



Now while the area of the first class of rectangles is the same 

 for all points of the curve, that of the second and of the squares 

 diminishes as E / and the efficiency increase from point to point 

 along the curve in the direction D, C, T, B, A. 



If the same analysis is applied to curve II it will be found 

 that the area of the rectangle expressing the electrical work 

 spent upon the motor per second increases, the rectangles deno- 

 ting the rate at which energy is supplied by the generator 

 all equal EW, and the area of the square representing the 

 waste in heat diminishes from point to point, as E, E, and the 

 efficiency increase. 



We may therefore distinguish three cases, E being constant 

 in each case : 



First, When E is constant. 



Then W ; is a maximum when E, is half E by Jacobi's law. 

 The electrical efficiency increases with E ; . 

 The current diminishes as E, increases. 



W diminishes from a maximum to zero when the efficiency is 

 unity. 



